On the number of unique expansions in non-integer bases

TL;DR

This paper investigates the set of numbers with unique expansions in non-integer bases, comparing their sizes for different bases with the same integer part, revealing properties of these unique representations.

Contribution

It analyzes the size and properties of the set of uniquely representable numbers in non-integer bases, especially comparing different bases with the same integer part.

Findings

The set of unique expansions varies with the base q.

Comparison of the size of unique expansion sets for different bases q and r.

Insights into the structure of numbers with unique expansions in non-integer bases.

Abstract

Let $q > 1$ be a real number and let $m=m(q)$ be the largest integer smaller than $q$. It is well known that each number $x \in J_q:=[0, \sum_{i=1}^{\infty} m q^{-i}]$ can be written as $x=\sum_{i=1}^{\infty}{c_i}q^{-i}$ with integer coefficients $0 \le c_i < q$. If $q$ is a non-integer, then almost every $x \in J_q$ has continuum many expansions of this form. In this note we consider some properties of the set $\mathcal{U}_q$ consisting of numbers $x \in J_q$ having a unique representation of this form. More specifically, we compare the size of the sets $\mathcal{U}_q$ and $\mathcal{U}_r$ for values $q$ and $r$ satisfying $1< q < r$ and $m(q)=m(r)$.